9/20/2018
Bayesian hierarchical model.
\(\begin{align*} [\mathbf{Z}, \boldsymbol{\theta}_D, \boldsymbol{\theta}_P | \mathbf{y}] & \propto [\mathbf{y} | \boldsymbol{\theta}_D, \mathbf{Z}] [\mathbf{Z} | \boldsymbol{\theta}_P] [\boldsymbol{\theta}_D] [\boldsymbol{\theta}_P] \end{align*}\)
Bayesian hierarchical model.
\(\begin{align*} \color{cyan}{[\mathbf{Z}, \boldsymbol{\theta}_D, \boldsymbol{\theta}_P | \mathbf{y}]} & \propto [\mathbf{y} | \boldsymbol{\theta}_D, \mathbf{Z}] [\mathbf{Z} | \boldsymbol{\theta}_P] [\boldsymbol{\theta}_D] [\boldsymbol{\theta}_P] \end{align*}\)
Posterior.
Bayesian hierarchical model.
\(\begin{align*} \color{cyan}{[\mathbf{Z}, \boldsymbol{\theta}_D, \boldsymbol{\theta}_P | \mathbf{y}]} & \propto \color{red}{[\mathbf{y} | \boldsymbol{\theta}_D, \mathbf{Z}]} [\mathbf{Z} | \boldsymbol{\theta}_P] [\boldsymbol{\theta}_D] [\boldsymbol{\theta}_P] \end{align*}\)
Posterior.
Data Model
Bayesian hierarchical model.
\(\begin{align*} \color{cyan}{[\mathbf{Z}, \boldsymbol{\theta}_D, \boldsymbol{\theta}_P | \mathbf{y}]} & \propto \color{red}{[\mathbf{y} | \boldsymbol{\theta}_D, \mathbf{Z}]} \color{blue}{[\mathbf{Z} | \boldsymbol{\theta}_P]} [\boldsymbol{\theta}_D] [\boldsymbol{\theta}_P] \end{align*}\)
Posterior.
Data Model.
Process Model.
Bayesian hierarchical model.
\(\begin{align*} \color{cyan}{[\mathbf{Z}, \boldsymbol{\theta}_D, \boldsymbol{\theta}_P | \mathbf{y}]} & \propto \color{red}{[\mathbf{y} | \boldsymbol{\theta}_D, \mathbf{Z}]} \color{blue}{[\mathbf{Z} | \boldsymbol{\theta}_P]} \color{orange}{[\boldsymbol{\theta}_D] [\boldsymbol{\theta}_P]} \end{align*}\)
Posterior.
Data Model.
Process Model.
Prior Model.
\(\begin{align*} [\mathbf{Z}, \boldsymbol{\theta}_D, \boldsymbol{\theta}_P | \mathbf{y}] & \propto \color{red}{[\mathbf{y} | \boldsymbol{\theta}_D, \mathbf{Z}]} [\mathbf{Z} | \boldsymbol{\theta}_P] [\boldsymbol{\theta}_D] [\boldsymbol{\theta}_P] \end{align*}\)
For location \(\mathbf{s}\) and time \(t\),
\(\begin{align*} \mathbf{y} \left( \mathbf{s}_i, t \right) & = \left( y_{1} \left( \mathbf{s}_i, t \right), \ldots, y_{d} \left( \mathbf{s}_i, t \right) \right)' \end{align*}\)
is an observation of a \(d\)-dimensional compositional count.
\(\begin{align*} \mathbf{y}\left( \mathbf{s}_i, t \right) | \mathbf{p}\left( \mathbf{s}_i, t \right) & \sim \operatorname{Multinomial} \left( N\left( \mathbf{s}_i, t \right), \mathbf{p}\left( \mathbf{s}_i, t \right) \right) \end{align*}\)
\(N\left( \mathbf{s}_i, t \right) = \sum_{j=1}^d y_{j}\left( \mathbf{s}_i, t \right)\) is the total count observed (fixed and known) for observation at location \(\mathbf{s}_i\) and time \(t\).
Compositional count vector \(\mathbf{y} \left( \mathbf{s}_i, t \right)\) a function of latent proportions \(\mathbf{p}\left( \mathbf{s}_i, t \right)\)
\(\begin{align*} \mathbf{p}\left( \mathbf{s}_i, t \right) | \boldsymbol{\alpha}\left( \mathbf{s}_i, t \right) & \sim \operatorname{Dirichlet} \left( \boldsymbol{\alpha}\left( \mathbf{s}_i, t \right) \right) \end{align*}\)
\(\begin{align*} \mathbf{y}\left( \mathbf{s}_i, t \right) | \boldsymbol{\alpha}\left( \mathbf{s}_i, t \right) & \sim \operatorname{Dirichlet-Multinomial} \left( N\left( \mathbf{s}_i, t \right), \boldsymbol{\alpha}\left( \mathbf{s}_i, t \right) \right) \end{align*}\)
\(\begin{align*} [\mathbf{Z}, \boldsymbol{\theta}_D, \boldsymbol{\theta}_P | \mathbf{y}] & \propto [\mathbf{y} | \boldsymbol{\theta}_D, \mathbf{Z}] \color{blue}{[\mathbf{Z} | \boldsymbol{\theta}_P]}[\boldsymbol{\theta}_D] [\boldsymbol{\theta}_P] \end{align*}\)
\(\begin{align*} \mathbf{z} \left(t \right) - \mathbf{X} \left( t \right) \boldsymbol{\gamma} & = \mathbf{M}\left(t\right) \left( \mathbf{z} \left(t-1 \right) - \mathbf{X} \left( t \right) \boldsymbol{\gamma} \right) + \boldsymbol{\eta} \left(t \right) \end{align*}\)